Theorem opid 3906

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Hypothesis

Ref	Expression
opid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
opid	⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 3708	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2238	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 3777	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		opid.1	. . 3 ⊢ 𝐴 ∈ V
5	4, 4	dfop 3887	. 2 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6		dfsn2 3708	. 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	3, 5, 6	3eqtr4i 2265	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1398   ∈ wcel 2205  Vcvv 2815  {csn 3694  {cpr 3695  ⟨cop 3697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703

This theorem is referenced by:  dmsnsnsng  5245  funopg  5391  funopsn  5865